Likelihood Ratio Test in Multivariate Linear Regression: from Low to High Dimension

Yinqiu He, Tiefeng Jiang, Jiyang Wen, Gongjun Xu

Research output: Contribution to journalArticlepeer-review


Multivariate linear regressions are widely used statistical tools in many applications to model the associations between multiple related responses and a set of predictors. To infer such associations, it is often of interest to test the structure of the regression coefficients matrix, and the likelihood ratio test (LRT) is one of the most popular approaches in practice. Despite its popularity, it is known that the classical χ2 approximations for LRTs often fail in high-dimensional settings, where the dimensions of responses and predictors (m, p) are allowed to grow with the sample size n. Though various corrected LRTs and other test statistics have been proposed in the literature, the important question of when the classic LRT starts to fail is less studied; an answer to this would provide insights for practitioners, especially when analyzing data with m/n and p/n small but not negligible. Moreover, the power performance of the LRT in high-dimensional data analysis remains underexplored. To address these issues, the first part of this work gives the asymptotic boundary where the classical LRT fails and develops the corrected limiting distribution of the LRT for a general asymptotic regime. The second part of this work further studies the test power of the LRT in the high-dimensional settings. The result not only advances the current understanding of asymptotic behavior of the LRT under alternative hypothesis, but also motivates the development of a power-enhanced LRT. The third part of this work considers the setting with p > n, where the LRT is not well-defined. We propose a two-step testing procedure by first performing dimension reduction and then applying the proposed LRT. Theoretical properties are developed to ensure the validity of the proposed method. Numerical studies are also presented to demonstrate its good performance.

Original languageEnglish (US)
JournalUnknown Journal
StatePublished - Dec 17 2018
Externally publishedYes


  • And phrases: High dimension
  • Likelihood ratio test
  • Multivariate linear regression

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